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4.8 – Tree diagrams

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DRILL:

There are 3 yellow and 2 green counters in a bag. I take two counters at random. Determine the probability that:

  1. They are of the same colour.
  2. They are of different colours.

 

 

a

b

 

The probability I hit a target on each shot is 0.3. I keep firing until I hit the target. Determine the probability I hit the target on the 5th shot.

 

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c

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DRILL:

There are 3 yellow and 2 green counters in a bag. I take two counters at random. Determine the probability that:

  • They are of the same colour.
  • They are of different colours.

 

 

a

b

 

The probability I hit a target on each shot is 0.3. I keep firing until I hit the target. Determine the probability I hit the target on the 5th shot.

 

c

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This problem follows the Geometric Probability Distribution, which models the number of trials until the first success in a sequence of independent Bernoulli trials (where each trial has two possible outcomes: success or failure).

The probability I hit a target on each shot is 0.3. I keep firing until I hit the target. Determine the probability I hit the target on the 5th shot.

Understanding the Problem

  • The probability of hitting the target on each shot is 0.3.
  • You keep firing until you hit the target.
  • You want to find the probability that you hit the target on the 5th shot.

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Trees are useful when you have later events conditioned on earlier ones, or in general when you have lots of conditional probabilities.

Example: You have two bags, the first with 5 red balls and 5 blue balls, and the second with 3 red balls and 6 blue balls. You first pick a ball from the first bag, and place it in the second. You then pick a ball from the second bag. Complete the tree diagram.

 

 

 

 

 

 

 

1st pick

2nd pick

 

 

 

 

 

 

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Trees are useful when you have later events conditioned on earlier ones, or in general when you have lots of conditional probabilities.

Example: You have two bags, the first with 5 red balls and 5 blue balls, and the second with 3 red balls and 6 blue balls. You first pick a ball from the first bag, and place it in the second. You then pick a ball from the second bag. Complete the tree diagram.

 

 

 

 

 

 

 

1st pick

2nd pick

 

 

 

 

 

 

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Key Point: When you need to find a probability using a tree, consider all possible paths in which that event is satisfied, and add the probabilities together.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Key Point: When you need to find a probability using a tree, consider all possible paths in which that event is satisfied, and add the probabilities together.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Of 120 competitors in a golf tournament, 68 reached the green with their tee shot on the first hole. Of these, 46 completed the hole in 3 shots or less. In total, 49 players took more than 3 shots on the first hole.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Click to reveal Tree Diagram

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Of 120 competitors in a golf tournament, 68 reached the green with their tee shot on the first hole. Of these, 46 completed the hole in 3 shots or less. In total, 49 players took more than 3 shots on the first hole.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Mini - Plenary

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Mini - Plenary

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P(H) = (5/12 x 2/3) + (7/12 x ½) = 41/72

​

P(R|H) = P(R n H) / P(H)

= (5/18) / (41/72)

= 20/41

​

P(RR or BB) = (5/12)2 + (7/12)2 = 37/72

b

c

d

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5

12

7

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P(H) = (5/12 x 2/3) + (7/12 x ½) = 41/72

​

P(R|H) = P(R n H) / P(H)

= (5/18) / (41/72)

= 20/41

​

P(RR or BB) = (5/12)2 + (7/12)2 = 37/72

b

c

d

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